diff --git a/src/functions.jl b/src/functions.jl
index 83f8380..509de30 100644
--- a/src/functions.jl
+++ b/src/functions.jl
@@ -12,13 +12,44 @@ function square(x::Real)
 end
 
 
-invpow2(x::Real, p::Integer) = sign(x) * abs(x)^inv(p)
-invpow2(x::Real, p::Real) = x ≥ zero(x) ? x^inv(p) : throw(DomainError(x, "inverse for x^$p is not defined at $x"))
-invpow2(x, p) = x^inv(p)
+function invpow2(x::Real, p::Integer)
+    if x ≥ zero(x) || isodd(p)
+        copysign(abs(x)^inv(p), x)
+    else
+        throw(DomainError(x, "inverse for x^$p is not defined at $x"))
+    end
+end
+function invpow2(x::Real, p::Real)
+    if x ≥ zero(x)
+        x^inv(p)
+    else
+        throw(DomainError(x, "inverse for x^$p is not defined at $x"))
+    end
+end
+function invpow2(x, p::Real)
+    # complex x^p is only invertible for p = 1/n
+    if isinteger(inv(p))
+        x^inv(p)
+    else
+        throw(DomainError(x, "inverse for x^$p is not defined at $x"))
+    end
+end
 
-invpow1(b, x) = log(abs(b), abs(x))
+function invpow1(b::Real, x::Real)
+    if b ≥ zero(b) && x ≥ zero(x)
+        log(b, x)
+    else
+        throw(DomainError(x, "inverse for $b^x is not defined at $x"))
+    end
+end
 
-invlog1(b::Real, x::Real) = b ≥ zero(b) && x ≥ zero(x) ? b^x : throw(DomainError(x, "inverse for log($b, x) is not defined at $x"))
+function invlog1(b::Real, x::Real)
+    if b ≥ zero(b)
+        b^x
+    else
+        throw(DomainError(x, "inverse for log($b, x) is not defined at $x"))
+    end
+end
 invlog1(b, x) = b^x
 
 invlog2(b, x) = x^inv(b)
diff --git a/test/test_inverse.jl b/test/test_inverse.jl
index 57fd92a..fe8599b 100644
--- a/test/test_inverse.jl
+++ b/test/test_inverse.jl
@@ -37,7 +37,7 @@ InverseFunctions.inverse(f::Bar) = Bar(inv(f.A))
     x = rand()
     for f in (
             foo, inv_foo, log, log2, log10, log1p, sqrt,
-            Base.Fix2(^, rand()), Base.Fix2(^, rand([-10:-1; 1:10])), Base.Fix1(^, rand()), Base.Fix1(log, rand()), Base.Fix2(log, rand()),
+            Base.Fix2(^, rand()), Base.Fix2(^, rand([-10:-1; 1:10])), Base.Fix1(^, rand()), Base.Fix1(log, rand()), Base.Fix1(log, 1/rand()), Base.Fix2(log, rand()),
         )
         InverseFunctions.test_inverse(f, x)
     end
@@ -55,10 +55,16 @@ InverseFunctions.inverse(f::Bar) = Bar(inv(f.A))
     # ensure that inverses have domains compatible with original functions
     @test_throws DomainError inverse(Base.Fix1(*, 0))
     @test_throws DomainError inverse(Base.Fix2(^, 0))
+    @test_throws DomainError inverse(Base.Fix1(log, -2))(5)
     @test_throws DomainError inverse(Base.Fix1(log, 2))(-5)
-    InverseFunctions.test_inverse(Base.Fix1(log, 2), -5 + 0im)
+    InverseFunctions.test_inverse(inverse(Base.Fix1(log, 2)), complex(-5))
     @test_throws DomainError inverse(Base.Fix2(^, 0.5))(-5)
-    InverseFunctions.test_inverse(Base.Fix2(^, 0.5), -5 + 0im)
+    @test_throws DomainError inverse(Base.Fix2(^, 0.51))(complex(-5))
+    InverseFunctions.test_inverse(Base.Fix2(^, 0.5), complex(-5))
+    @test_throws DomainError inverse(Base.Fix2(^, 2))(-5)
+    @test_throws DomainError inverse(Base.Fix1(^, 2))(-5)
+    @test_throws DomainError inverse(Base.Fix1(^, -2))(3)
+    @test_throws DomainError inverse(Base.Fix1(^, -2))(3)
 
     A = rand(5, 5)
     for f in (