diff --git a/doc/api/deprecations.md b/doc/api/deprecations.md
index c70f494c6ddada..d0754285a4f6e4 100644
--- a/doc/api/deprecations.md
+++ b/doc/api/deprecations.md
@@ -700,6 +700,16 @@ Type: Runtime
 
 `REPLServer.turnOffEditorMode()` was removed from userland visibility.
 
+<a id="DEP0079"></a>
+### DEP0079: Custom inspection function on Objects via .inspect()
+
+Type: Documentation-only
+
+Using a property named `inspect` on an object to specify a custom inspection
+function for [`util.inspect()`][] is deprecated. Use [`util.inspect.custom`][]
+instead. For backwards compatibility with Node.js prior to version 6.4.0, both
+may be specified.
+
 [`Buffer.allocUnsafeSlow(size)`]: buffer.html#buffer_class_method_buffer_allocunsafeslow_size
 [`Buffer.from(array)`]: buffer.html#buffer_class_method_buffer_from_array
 [`Buffer.from(buffer)`]: buffer.html#buffer_class_method_buffer_from_buffer
@@ -738,6 +748,8 @@ Type: Runtime
 [`util._extend()`]: util.html#util_util_extend_target_source
 [`util.debug()`]: util.html#util_util_debug_string
 [`util.error()`]: util.html#util_util_error_strings
+[`util.inspect()`]: util.html#util_util_inspect_object_options
+[`util.inspect.custom`]: util.html#util_util_inspect_custom
 [`util.isArray()`]: util.html#util_util_isarray_object
 [`util.isBoolean()`]: util.html#util_util_isboolean_object
 [`util.isBuffer()`]: util.html#util_util_isbuffer_object
diff --git a/doc/api/util.md b/doc/api/util.md
index ce56c50104dbc4..2653c01af1d4c9 100644
--- a/doc/api/util.md
+++ b/doc/api/util.md
@@ -377,8 +377,8 @@ terminals.
 <!-- type=misc -->
 
 Objects may also define their own `[util.inspect.custom](depth, opts)`
-(or, equivalently `inspect(depth, opts)`) function that `util.inspect()` will
-invoke and use the result of when inspecting the object:
+(or the equivalent but deprecated `inspect(depth, opts)`) function that
+`util.inspect()` will invoke and use the result of when inspecting the object:
 
 ```js
 const util = require('util');
@@ -388,7 +388,7 @@ class Box {
     this.value = value;
   }
 
-  inspect(depth, options) {
+  [util.inspect.custom](depth, options) {
     if (depth < 0) {
       return options.stylize('[Box]', 'special');
     }
@@ -427,21 +427,6 @@ util.inspect(obj);
 // Returns: "{ bar: 'baz' }"
 ```
 
-A custom inspection method can alternatively be provided by exposing
-an `inspect(depth, opts)` method on the object:
-
-```js
-const util = require('util');
-
-const obj = { foo: 'this will not show up in the inspect() output' };
-obj.inspect = function(depth) {
-  return { bar: 'baz' };
-};
-
-util.inspect(obj);
-// Returns: "{ bar: 'baz' }"
-```
-
 ### util.inspect.custom
 <!-- YAML
 added: v6.6.0